Enotes requires that only one question be asked at a time; therefore, your question has been edited to reflect this.
The relationship between Lady Capulet (Juliet's mom) and Juliet's cousin Tybalt is one of the more controversial aspects of this version, which is actually called Romeo + Juliet and stars Leonardo DiCaprio and Claire Danes. The ironic twists in this version, such as the relationship you mention, are precisely what I think make this version so unique.
Shakespeare doesn't really give an indication of an incestuous relationship between Lady Cauplet and Tybalt. Of course, there's nothing that says there isn't an incestuous relationship either. That's what makes artistic licence so very delicious in regards to film.
In this version, Lady Capulet definitely does have some kind of sexual relationship with Tybalt. There is a shred of evidence within the very first moment you see the two together (at the Capulet party). You can't escape it. The two are passionately kissing each other. Lady Capulet is dressed as Cleopatra and Tybalt is aptly dressed as the devil. (It is a costume party after all.) In there defense, there are definitely a lot of alcohol and drugs running rampant at this party; therefore, the two could truly be under the influence and not know what they are doing. Somehow, I doubt it, though.
Because there is a incestuous, sexual relationship at work here, Lady Capulet will, no doubt, be greatly affected (even more so than in the original play) when Tybalt is killed by Romeo. Where there may have been some compassion towards Juliet who has finally found her one true love in Romeo, now Lady Capulet will have none. Romeo has killed her nephew, ... and her lover.
Thus, this relationship in addition to adding to the twisted artistic licence of Romeo + Juliet (as compared to Shakespeare's Romeo and Juliet), adds to the degree of intensity regarding Lady Capulet's reaction to Tybalt's death and increases the vast ire towards Romeo for his actions. Could there ever be a story with more woe?